Maximum Precision in Elliott-Donaldson Formulae

This paper considers the approximation of by quadratures which are constructed inthe manner described by Elliott & Donaldson. Results areestablished for choosing the nodes so that the quadrature is exact on as wide a range of positive and negativepowers of z as possible.     

On the Determination of Quadrature Formulae of Highest Degree of Precision for Approximating Fourier Coefficients

Let Q₀(x), Q₁(x),..., Q_n(x),... be a sequence of polynomialswhich are orthogonal with respect to the inner product . In estimating the Fourier coefficients a₁ = $$\langlef,{Q}_{i}\rangle $$, it is natural to use a quadrature formulaof highest possible degree of precision to approximate $${\int}_{a}^{b}W\left(x\right)f\left(x\right)dx$$ where W(x) = w(x)Q_i(x).Since the weight function W(x) changes sign i times in (a, b),the usual results of quadrature theory do not apply. This paperdevelops a procedure which is an initial attempt to determinewhat degree of precision is attainable.     

Closed-form Representations for Errors of Cubic Spline Approximations on Equally Spaced Knots

This paper considers estimating errors in the approximationof an arbitrary linear functional on C⁴[0, L] by use of cubicsplines on equally spaced knots. Using explicit formulas derivedfor the cubic spline approximations of f_k(x) = cos (kx/L), formulasare found for the cosine expansion of the Peano Kernel for theremainder functional.